How many schedules are there for the following:

Question

Jo is going on a 8 day activities holiday, each day she can choose 1 activity from 3, which are, kayaking, sailing and land-based sports. Jo never does different water sports on consecutive days, she also wants to try all the activities at least once in her 8 day holiday.

Bear in mind that she can also decide not to do anything for a day adding another possibility.

Answer 1

Let 1 = kayaking, 2 = sailing , 3 = land-based sports, and 4 = none

The easiest way to do this problem is to write 8 loops 1-4 and count all cases where the adjacent loops don't add up to 3 (the multiplication in the code accomplishes this) and each of 1-3 occurs at least once (the z vector accomplishes this).

I got a total count of 14894

The R code is below. The tempfile written out will allow you to check that the code worked.

count <- 0

tempfile <- c(a=0,b=0,c=0,d=0,e=0,f=0,g=0,h=0)

for(a in 1:4){

for(b in 1:4){

for(c in 1:4){

for(d in 1:4){

for(e in 1:4){

for(f in 1:4){

for(g in 1:4){

for(h in 1:4){

#check if no adjacent different water sports

if((a+b-3)*(b+c-3)*(c+d-3)*(d+e-3)*(e+f-3)*(f+g-3)*(g+h-3) != 0){

z <- rep(0,times=3)

y <- c(a,b,c,d,e,f,g,h)

for(loop in 1:8){if(y[loop] != 4){z[y[loop]] <- 1}}

#check that all sports are included at least once

if(z[1]*z[2]*z[3] > 0){

tempfile <- rbind(tempfile,y)

count <- count + 1}}

}}}}}}}}

print(count)

tempfile <- data.frame(tempfile)

tempfile <- tempfile[-1,]

write.csv(tempfile,row.names=FALSE)